where \(\bar{{\bf V}}(\Lambda )\)  and \(\widetilde{{\bf V}}(\Lambda )\)  are \((n-1)\times (n-1)\)  matrices
obtained by deleting the first row and column from the Vandermonde matrix \({\bf V}(\Lambda )\)  and its
conjugate transpose, separately. The \((i,j)\) -th entry of \({\bf V}(\Lambda )\)  and its conjugate transpose
are \(\lambda _i^{j-1}\)  and \(\lambda _j^{i-1}\) , respectively. Thanks to [1]}, the
result in (REF ) can be obtained through integration over \(n-1\)  eigenvalues from
\(\lambda _2\)  to \(\lambda _n\) . The final expressions are in both cases due to the definition of the
scalar Beta function [2]}. It should be noticed that the choice of \(\lambda _1\) 
in (REF ) has no effect on the final result, since we started from an unordered
eigenvalue distribution. Using the same approach, the proof of (REF ) is
straightforward.
